# Ohm’s law holding true on temperature-dependent resistances

As far as I know, Ohm's law (in macroscopic form) states that the in some devices/conductors/materials, the (instantaneous) current through the device is directly proportional to the (instantaneous) voltage across the device. It can be proven this occurs only if the (static/DC) resistance of the device is constant; if it is variable, then the voltage and current aren't directly proportional, so Ohm's law is not satisfied.

In the above, I haven’t discussed about temperature. In metallic conductors, as we know their resistivity is dependent on the temperature of the conductor, which depends on the ambient temperature and the current flowing through the conductor (Joule’s law). Since the resistance depends on the resistivity, it follows the resistance of a metallic conductor depends on the current through it. Doesn’t that make metallic conductors not obey Ohm’s law (since the voltage is not directly proportional to current)? Or am I wrong (if so, how)?

(This question was originally longer, but people suggested to shorten it, so I did it. The content of the original post is in this Quora answer.)

• No material has one unique resistance across all parameters. Nov 4 '21 at 22:55
• And the reason for the downvote is? I did my research on the topic, both on knowledge and on looking for similar questions on this site. Nov 4 '21 at 22:57
• @Alejandro Nava Perfectly sound discussions (like yours) that distinguish between $V=IR$ as a definite of resistance and $V=IR$ as a law of nature if $R$ is independent of $I$ (or $V$) can generate angry responses - and downvotes. I, too, have had them. Nov 5 '21 at 0:07
• (Too)much ado about (almost) nothing. Definitions and labels mostly. But I did not downvote.
– nasu
Nov 5 '21 at 1:56
• @Alejandro Nava These are all well known points and the differential resistance is defined to take into account the bhavior of nonlinear devices. If the relationship between current and tension is a law or not or which version should be called "Ohm's law" is just a matter of convention or labeling. Hooke's law have the same "problem". Real springs may have spring "constants" dependent on the extensions. All the "laws" that assume a linear behavior have this issue. So what? We know that they don't work for nonlinear behavior.
– nasu
Nov 5 '21 at 16:13

There seem to be at least two valid ways in which physicists and engineers use the term Ohm's law, neither of which is merely a definition of resistance.

(a) If $$I$$ is proportional to $$V$$ then the conductor obeys Ohm's law, otherwise it doesn't.

(b) At constant temperature (and, strictly, at constant pressure) metallic conductors and most single-substance conductors exhibit $$I$$ proportional to $$V$$.

Clearly Ohm's law as defined in (a) is obeyed only by a narrow class of conductors. But (b) attempts to be a self-contained law of nature with few exceptions.

Your thermistor doesn't obey Ohm's law according to (a), for the reasons that you state.

On the other hand, (b) has nothing to say on whether or not a thermistor, self-heating or externally heated, obeys Ohm's law. This is because (b) specifically doesn't deal with conductors that aren't at constant temperature!

My own preference is for (a), but as nasu implies (in his comment on the question) the matter is of no great moment: you can usually work out from the context what someone means by Ohm's law.

• No down vote here! $V=R(T)I$. $R(T)$ takes many shapes, from (relatively) constant to complicated, and perhaps not even a function at all. In a manner of speaking, solid state physics is all about understanding $R(T)$. Nov 5 '21 at 1:34
• By any chance, do you know what Georg Ohm referred to in his book? Was it (a) or (b)? Nov 5 '21 at 15:26
• Florian Cajori's A History of Physics, some 90 years's old but still, I believe, available as a Dover book, has a few pages on Ohm. Ohm used thermo-electric emfs and a home-made galvanometer to establish that the current through a metal wire was proportional to the applied voltage. [He allowed for a constant series resistance due to things in the circuit other than the wire.] Cajori doesn't suggest that Ohm noticed any temperature effects, or tried anything but metal wires as conductors. So I suspect that Ohm's version of his law was more like (a) than (b). Nov 5 '21 at 16:13

The recovering ex-engineer weighs in.

Note that for most common engineering metals, the temperature coefficient of resistance is small. This means that for temperatures not far from ambient, the shift in resistance caused by ohmic heating is small enough to ignore, and so engineers (even recovering ex-engineers) will indeed ignore it.

This means that engineers will use Ohm's law as a law and not just as a definition of resistance. A further reason for this practice is as follows:

In the field of dynamic systems modeling, an Ohmic (i.e., linear) dependence of a flow variable on a resistive element furnishes a satisfactory accounting for the behavior of electrical, mechanical, hydraulic and thermodynamic systems subject to small displacements- certainly good enough for first-order models to yield sufficiently accurate predictions (two-place accuracy) of system behavior for those models to be useful.

In general and in my experience, three-place accuracy requires accounting for second-order effects- and the dynamic systems model of a temperature-dependent resistance will replace a single-valued function with a value pulled from a lookup table that contains the (precalculated) second-order dynamics.

Even this is insufficient to prevent engineers and recovering ex-engineers from considering ohmic behavior as a law and not just a definition of resistance.